Exploratory Social Network Analysis with Pajek
 W. de Nooy, A. Mrvar, V. Batagelj
Part I - Fundamentals
Social network analysis focuses on relations between people, groups of people,
organizations, countries, et cetera. These relations combine into networks which
we will learn to analyze.
In the first part of the book, we introduce the concept of a social network. We
discuss several types of networks and the ways in which we can analyze them
numerically and visually with the computer software (program Pajek) which is
used throughout this book.
After studying Chapters 1 and 2, you should understand the concept of a
social network and you should be able to create, manipulate, and visualize a
social network with the software provided with this book.
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Exploratory Social Network Analysis with Pajek
 W. de Nooy, A. Mrvar, V. Batagelj
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Exploratory Social Network Analysis with Pajek
 W. de Nooy, A. Mrvar, V. Batagelj
1 Looking for social structure
1.1
Introduction
The social sciences focus on structure: the structure of human groups,
communities, organizations, markets, society, or the world system. In this book,
we conceptualize social structure as a network of social relations. Social network
analysts assume that interpersonal relations matter, as do relations between
organizations or countries, because they transmit behavior, attitudes, information,
or goods. Social network analysis offers the methodology to analyze social
relations. It tells us how to conceptualize social networks and how to analyze
them.
In this book, we will present the most important methods of exploring social
networks, emphasizing visual exploration of social networks. Network
visualization has been an important tool for researchers from the very beginning
of social network analysis. This chapter introduces the basic elements of a social
network and shows how to construct and draw a social network.
1.2
Sociometry and sociogram
The basis of social network visualization was laid by researchers who called
themselves sociometrists. Their leader, J.L. Moreno, founded a social science
called sociometry, which studied interpersonal relations. Society, it was argued, is
not an aggregate of individuals and their characteristics, as statisticians assume,
but a structure of interpersonal relations. Therefore, the individual is not the basic
social unit. The social atom consists of an individual and his or her social,
economic, or cultural relations. Social atoms are linked into groups, and,
ultimately, society consists of interrelated groups.
From their point of view, it is understandable that sociometrists studied the
structure of small groups rather than the structure of society at large. In particular,
they investigated social choices within a small group. They asked people
questions like: Whom would you choose as a friend, colleague, advisor, etc.? This
type of data has since been known as sociometric choice. In sociometry, social
choices are considered to be the most important expression of social relations.
Figure 1 presents an example of sociometric research. It depicts the choices of 26
girls living in one ‘cottage’ (dormitory) at a New York State Training School.
The girls were asked to choose the girls they liked best as their dining-table
partners. First and second choices are selected only. The data can be found in the
file Dining-table_partners.net on the CD ROM.
3
 W. de Nooy, A. Mrvar, V. Batagelj
Exploratory Social Network Analysis with Pajek
Louise
2
Ada
2
Lena
1
1
1
2
1
Marion
2
Cora
1
2
2
Frances 1
Maxine
1
1
Jean
2
1
1
2
1
Laura
Ruth
Anna
1
Betty 2
2
1
2
2
1
Hazel
Helen
Ellen
2
1
1
2
2
Alice
Mary
2
Edna
1
Martha
Robin
Jane
2
2
1
2
1
1
2 2
Eva
Adele
2
Hilda
1
1
Ella
2
1
Irene
Figure 1 - Sociogram of dining-table partners.
Figure 1 is an example of a sociogram, which is a graphical representation of
group structure. The sociogram is among the most important instruments
originated in sociometry and it is the basis for the visualization of social
networks. Probably, you have already ‘read’ and understood the figure without
the explanation that follows now, which illustrates its visual appeal and
conceptual clarity. In the sociogram, each girl of the dormitory is represented by a
circle. In this network, girls are the social actors. For the sake of identification,
names are written next to the circles.
Each arc represents a choice. The girl that chooses a peer as a dining-table
companion sends an arc towards her. In this sociogram, some mutual choices are
represented by blue lines because a single line is visually more attractive than a
pair of arcs. Cora and Ada (top left), for instance, choose each other as their
favorite dining-table partner, so their choices are replaced by a blue line with
value one. In contrast, Hilda and Hazel (bottom right) are not linked by equivalent
choices: Hilda is Hazel’s first choice but Hazel is Hilda’s second choice. Unequal
mutual choices are represented by two arcs, whereas equivalent mutual choices
are represented by a line. As a rule, this will be done throughout the book.
A sociogram depicts the structure of relations within a group. It shows which
girls are popular as indicated by the number of choices they receive, but also
whether the choices come from popular or unpopular girls. For example, Hilda
receives four choices from Irene, Ruth, Hazel, and Betty, and she reciprocates the
last two choices. But none of these four girls receives any choices. Therefore,
Hilda is located at the margin of the sociogram, whereas Frances, who is chosen
only twice, is more central because she is chosen by ‘popular’ girls like Adele
and Marion. A simple count of choices does not reveal this, but a sociogram does.
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Exploratory Social Network Analysis with Pajek
 W. de Nooy, A. Mrvar, V. Batagelj
The sociogram has proved to be an important analytic tool that helped to
reveal several structural features of social groups. In this book, we will make
ample use of it.
1.3
Exploratory social network analysis
Sociometry is not the only tradition in the social sciences that focuses on social
relations. Actually, the concept of a social network is attributed to the
anthropologist J. Barnes in the 1950s. Anthropologists study kinship relations,
friendship, and gift-giving among people rather than sociometric choice.
Furthermore, social psychologists focus on affections, political scientists study
power relations among people, organizations, or nations, and economists
investigate trade and organizational relations between firms. In this book, the
word actor refers to a person, organization, or nation which is involved in a
social relation. We may say that social network analysis studies the social
relations between actors.
The main goal of social network analysis is detecting and interpreting patterns of
social relations between actors.
This book deals with exploratory social network analysis only. This means that
we do not have specific hypotheses about the structure of a network beforehand,
which we can test. For example, a hypothesis on the dining-table partners
network could predict a particular rate of mutual choices, e.g., one out of five
choices will be reciprocated. This hypothesis must be grounded in social theory
and prior research experience. The hypothesis can be tested provided that an
adequate statistical model is available.
We will not use hypothesis testing here, because we cannot assume prior
research experience in a course book and because the statistical models involved
are complicated. Therefore, we adopt an exploratory approach which assumes
that the structure or pattern of relations in a social network is meaningful to the
members of the network and, hence, to the researcher. Instead of testing prespecified structural hypotheses, we will explore social networks for meaningful
patterns.
For similar reasons, we will not pay attention to the estimation of network
features from samples. In network analysis, estimation techniques are even more
complicated than estimation in statistics, since the structure of a random sample
seldom matches the structure of the overall network. It is easy to demonstrate
this. For example, select five girls from the dining-table partners network at
random and focus on the choices among them . You will find less choices per
person than the two choices in the overall network for the simple reason that
choices to girls outside the sample are neglected. Even in this simple respect, a
sample is not representative of a network.
Instead of samples, we will analyze entire networks. However, what is the
entire network? Sociometry assumed that society consists of interrelated groups,
so a network encompasses society at large. Research on the so-called Small
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Exploratory Social Network Analysis with Pajek
 W. de Nooy, A. Mrvar, V. Batagelj
World problem showed that ties of acquaintanceship connect us to almost every
human being on the earth in six or seven steps, i.e. with five or six intermediaries,
so our network eventually covers the entire world population, which is clearly too
large a network to be studied. Therefore, we must use an artificial criterion to
delimit the network which we will study. For example, we may study the girls of
one dormitory only. We do not know their preferences for table partners in other
dormitories. Maybe Hilda is the only vegetarian in a group of carnivores and she
prefers to eat with girls of other dormitories. If so, including choices between
members of different dormitories will alter Hilda’s position in the network
tremendously.
Since boundary specification may seriously affect the structure of a network,
it is important to consider it carefully. Use substantive arguments to support your
decision on whom to include in the network and whom to exclude.
Exploratory social network analysis consists of four parts: the definition of a
network, network manipulation, determination of structural features, and visual
inspection. In the following subsections we present an overview of these
techniques. This overview serves to introduce basic concepts in network analysis
and to help you getting started with the software which accompanies this book.
1.3.1
Network definition
In order to analyze a network, we must first have one. What is a network? Here,
and elsewhere, we will use a branch of mathematics called graph theory to define
concepts. Most characteristics of networks that we will introduce in this book
originate from graph theory. Although this is not a course in graph theory, you
should study the definitions carefully in order to understand what you are doing
when you apply network analysis. We present definitions in text boxes to
highlight them.
A graph is a set of vertices and a set of lines between pairs of vertices.
What is a graph? A graph represents the structure of a network and all it needs for
this is a set of vertices (which are also called points or nodes) and a set of lines
where each line connects two vertices.
A vertex (singular of vertices) is the smallest unit in a network. In social
network analysis, it represents an actor, e.g., a girl in the dormitory, an
organization or a country. A vertex is usually identified by a number.
A line is a relation between two vertices in a network. In social network
analysis it can be any social relation. A line is defined by its two endpoints,
which are the two vertices that are incident with the line.
A loop is a special kind of line, namely a line which connects a vertex to
itself. In the dining-table partners network, loops do not occur because girls are
not allowed to choose themselves as a dinner-table partner. We will find out,
however, that loops are meaningful in some kinds of networks.
A line is directed or undirected. A directed line is called an arc, whereas an
undirected line is an edge. Sociometric choice is best represented by arcs,
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Exploratory Social Network Analysis with Pajek
 W. de Nooy, A. Mrvar, V. Batagelj
because one girl chooses another and choices need not be reciprocated, e.g., Ella
and Ellen in Figure 1.
A directed graph or digraph contains one or more arcs. A social relation
which is undirected, e.g., is sister of, is represented by an edge because both
individuals are equally involved in the relation. An undirected graph does not
contain arcs: all of its lines are edges.
Formally, an arc is an ordered pair of vertices in which the first vertex is the
sender (the tail of the arc) and the second the receiver of the relation (the head of
the arc). An arc points from a sender to a receiver. In contrast, an edge, which has
no direction, is represented by an unordered pair. It does not matter which vertex
is first or second in the pair. We should note, however, that an edge may usually
be replaced by a bi-directional arc: if Ella and Ellen are sisters (undirected), we
may say that Ella is the sister of Ellen and Ellen is the sister of Ella (directed). It
is important to note this, as we will see in later chapters.
The dining-table partners network has no multiple lines since a girl was not
allowed to nominate the same girl as first and second choice. Without this
restriction, which was imposed by the researcher, multiple arcs could have
occurred and they actually do occur in several social networks.
In a graph, multiple lines are allowed but when we say that a graph is simple,
we indicate that it has no multiple lines. In addition, a simple undirected graph
does not contain loops, whereas loops are allowed in a simple directed graph. It is
important to remember this.
A simple undirected graph contains neither multiple edges nor loops.
A simple directed graph does not contain multiple arcs.
Now that we have discussed the concept of a graph at some length, it is very easy
to define a network. A network consists of a graph and additional information on
the vertices or lines of the graph. We should note that the additional information
is irrelevant to the structure of the network because the structure depends on the
pattern of relations.
In the dining-table partners network, the names of the girls represent
additional information on the vertices which turns the graph into a network.
Thanks to this information, we can tell Ella from Ellen in the sociogram. The
numbers which are printed near the arcs and edges offer additional information
on the links between the girls: a one indicates a first choice and a two represents a
second choice. They are called line values and they usually indicate the strength
of a relation.
A network consists of a graph and additional information on the vertices or the
lines of the graph.
The dining-table partners network is clearly a network and not a graph. It is a
directed simple network because it contains arcs (directed) but not multiple arcs
(simple). In addition, we know that it does not contain loops. Several analytic
techniques which we will discuss assume that loops and multiple lines are absent
7
Exploratory Social Network Analysis with Pajek
 W. de Nooy, A. Mrvar, V. Batagelj
from a network. However, we will not spell out these properties of each network
but we will simply indicate whether it is simple or not. Take care!
Application
network data file
In this book, we will learn social network analysis by doing it. We will use the
computer program Pajek - Slovenian for spider - to analyze and draw social
networks. The software can be installed on your computer from the CD ROM
which accompanies this book (see Appendix 1).
Some concepts from graph theory are the ‘building blocks’ or ‘data objects’
of Pajek. Of course, a network is the most important data object in Pajek, so let us
describe it first. In Pajek, a network is defined in accordance with graph theory: a
list of vertices and lists of arcs and edges, where each arc or edge has a value.
Take a look at the partial listing of the data file for the dining-table partners
network (Figure 2, note that part of the vertices and arcs are replaced by […]).
Open the file Dining-table_partners.net in a word processor to see the
entire data file.
First, the data file specifies the number of vertices. Then, each vertex is
identified on a separate line by a serial number, a textual label (enclosed in “”)
and three real numbers between 0 and 1, which indicate the position of the vertex
in 3-dimensional space if the network is drawn. We will pay more attention to
these coordinates in Chapter 2. For now, it suffices to know that the first number
specifies the horizontal position of a vertex (0 is at the left of the screen and 1 at
the right) and the second number gives the vertical position of a vertex (0 is the
top of the screen, 1 is the bottom). The text label is crucial for identification of
vertices, the more so because serial numbers of vertices may change during the
analysis.
*Vertices 26
1 "Ada"
2 "Cora"
3 "Louise"
4 "Jean"
[…]
25 "Laura"
26 "Irene"
*Arcs
1
3
2
1
2
1
2
1
1
2
4
2
3
9
1
3 11
2
[…]
25 15
1
25 17
2
26 13
1
26 24
2
*Edges
0.1646
0.0481
0.3472
0.1063
0.1077
0.3446
0.0759
0.6284
0.5000
0.5000
0.5000
0.5000
0.5101 0.6557 0.5000
0.7478 0.9241 0.5000
Figure 2 - Partial listing of a network data file for Pajek.
The list of vertices is followed by a list of arcs. Each line identifies an arc by the
serial number of the sending vertex, followed by the number of the receiving
vertex and the value of the arc. Just like graph theory, Pajek defines a line as a
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Exploratory Social Network Analysis with Pajek
 W. de Nooy, A. Mrvar, V. Batagelj
pair of vertices. In Figure 2, the first arc represents Ada’s choice of Louise as a
dining-table partner. This is her second choice; Cora is her first choice, which is
indicated by the second arc. A list of edges is similar to a list of arcs with the
exception that the order of the two vertices which identify an edge is disregarded
in computations. In this data file, no edges are listed.
It is interesting to note that we can distinguish between the structural data or
graph and the additional information on vertices and lines in the network data
file. The graph is fully defined by the list of vertex numbers and the list of pairs
of vertices, which defines its arcs and edges. This part of the data, which is
printed in regular typeface in Figure 2, represents the structure of the network.
The vertex labels, coordinates, and line values (in italics) specify the additional
properties of vertices and lines which make these data a network. Although this
information is extremely useful, it is not required: Pajek will use vertex numbers
as default labels and set line values to one if they are not specified in the data file.
In addition, Pajek can use several other data formats, which we will not discuss.
They are listed in Appendix 1.
Figure 3 - Pajek Main screen.
File>Read
We will explain how to create a new network later. Let us first have a look at the
network of the dining-table partners. First, start Pajek by double-clicking the file
Pajek.exe on your hard disk. The computer will display the Main screen of
Pajek (Figure 3). From this screen, you can open the dining-table partners
network with the Read command in the File menu or by clicking the button with
an icon of a folder under the word Network. In both cases, the usual Windows file
dialog box appears with which you can search and select the file Dining-table
partners.net from the \Chapter1 directory on the CD ROM.
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Exploratory Social Network Analysis with Pajek
Network drop list
 W. de Nooy, A. Mrvar, V. Batagelj
When a network is read by Pajek, its name is displayed in the Network drop
list. This drop list is a list of the networks that are accessible to Pajek. You can
open a drop list by left-clicking on the button with the triangle at the right. The
network that you select in the list is shown when the list is closed, e.g., the
network Dining-table partners.net in Figure 3. Notice that the number of
vertices in the network is displayed in parentheses next to the name. The selected
network is the active network, meaning that any operation you perform on a
network will use this particular network. For example, if you use the Draw menu
now, Pajek draws the dining-partners network for you.
The Main screen displays five more drop lists beneath the Network drop list.
Each of these drop lists represents a data object in Pajek: partitions, permutations,
clusters, hierarchies, and vectors. Later chapters will familiarize you with these
data objects. Just note that each object can be opened, saved or edited from the
File menu or using the three icons left of a drop list (see section 1.4).
We advise you to open the network in Pajek and carry out the commands
which we will discuss in subsequent sections. This will familiarize you with the
concepts discussed as well as with Pajek.
1.3.2
Manipulation
In social network analysis, it is often useful to modify a network. For instance,
large networks are too big to be drawn, so we extract a meaningful part of the
network that we inspect first. Visualizations work much better for small (some
dozens of vertices) to medium sized (some hundreds of vertices) networks than
for large networks with thousands of vertices. When social networks contain
different kinds of relations, we may focus on one relation only, for instance, we
may want to study first choices only in the dining-table partners network. Finally,
some analytic procedures demand that complex networks with loops or multiple
lines are reduced to simple graphs first.
Application
menu structure
Network manipulation is a very powerful tool in social network analysis. In this
book, we will encounter several techniques to modify a network or select a
subnetwork. Network manipulation always results in a new network. In general,
many commands in Pajek produce new networks or other data objects rather than
graphical or tabular output. This motivates the central place of the data object
drop lists in the main menu of Pajek.
The commands for manipulating networks are accessible from menus in the
Main screen. The Main screen menus have a clear logic. Manipulations that
involve one type of data object are listed under a menu with the object’s name,
e.g., the Net menu contains all commands that operate on one network and the
Nets menu lists operations on two networks. Manipulations that need different
kinds of objects are listed in the Operations menu. When you try to locate a
command in Pajek, just consider which data objects you want to use.
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Exploratory Social Network Analysis with Pajek
Net>Transform
>Arcs→Edges
>Bidirected only
>Min Value
 W. de Nooy, A. Mrvar, V. Batagelj
An example may illustrate this. Suppose we want to change reciprocated
choices in the dining-table partners network into edges. Because this operation
concerns one network and no other data objects, we must look for it in the Net
menu. If we left-click on the word Net in the upper left of the Main screen, a drop
down menu is displayed. Position the cursor on the word Transform in the drop
down menu and a new submenu is opened with a command to change arcs into
edges (Arcs→Edges). Finally, we reach the option allowing us to change bidirectional arcs into edges and to assign the smallest line value of the two arcs to
the new edge which will replace them (see Figure 4). In this book, we will
abbreviate this sequence of commands to:
[Main]Net>Transform>Arcs→Edges>Bidirected only>Min Value.
The screen or window that contains the menu is presented between square
brackets and transition to a submenu is indicated by > . The screen name is
specified only if the context is ambiguous. The abbreviated command is also
displayed in the margin (see above) for the purpose of quick reference.
Figure 4 - Menu structure in Pajek.
When the command to change arcs into edges is executed, an information box
appears asking whether a new network must be made (Figure 5). If the answer is
yes, a new network named Bidirected Arcs to Edges of N1 (26) is
added to the Network drop list with serial number 2. On the other hand,
answering no to the question in the information box causes Pajek to change the
original network. This example highlights the use of menus in Pajek and their
notation in this book.
Figure 5 - An information box in Pajek.
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Exploratory Social Network Analysis with Pajek
1.3.3
 W. de Nooy, A. Mrvar, V. Batagelj
Calculation
In social network analysis, several structural features have been quantified, e.g.
an index that measures the centrality of a vertex. Some measures pertain to the
entire network, whereas others summarize the structural position of a subnetwork
or a single vertex. Calculation outputs a single number in the case of a network
characteristic and a series of numbers in the case of subnetworks and vertices.
Exploring network structure by calculation is much more concise and precise
than visual inspection. However, structural indices are sometimes abstract and
difficult to interpret. Therefore, we will use both visual inspection of a network
and calculation of structural indices to analyze network structure.
Application
Report screen
In Pajek, results of calculations are reported in a separate window which we will
call the Report screen. This screen displays numeric results that summarize
structural features as a single number, a frequency distribution, or a crosstabulation. Calculations that assign a value to each vertex are not reported in this
screen. They are stored as data objects in Pajek, notably partitions and vectors
(see Chapter 2). The Report screen displays text but no network drawings. The
contents of the Report screen can be saved as a text file from its File menu.
Figure 6 - Report screen in Pajek.
Info>Network>General
The Report screen depicted in Figure 6 shows the number of vertices, edges, and
arcs in the original dining-table partners network. This is general information on
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Exploratory Social Network Analysis with Pajek
 W. de Nooy, A. Mrvar, V. Batagelj
the network that is provided by the command Info>Network>General (as you
know now, this means the General command within the Network submenu of the
Info menu). Besides the number of vertices, edges, and arcs, the screen also
shows the number of loops and two indices of network density that will be
explained in Chapter 3. Also, this command displays the number of lines
requested in a dialog box which opens when the Info command is selected (Figure
7).
Figure 7 - Dialog box of Info>Network>General command.
In this example, we typed 10 in the dialog box, so the report screen shows 10
lines with the highest line values, which are second choices in the dining-table
partners network. In the report screen, each line is described by its rank according
to its line value, a pair of vertex numbers, the line value, and a pair of vertex
labels. Hence, the first line in Figure 6 represents the arc from Ada (vertex
number 1) to Louise (vertex number 3), which is Ada’s second choice. To get a
list of all first choices in the dining-table partners network, we should have typed
-26 or the range 27 52, which contains the arcs on ranks 27 up to and including
52, in the dialog box.
1.3.4
Visualization
The human eye is trained in pattern recognition. Therefore, network
visualizations help to trace and present patterns of relations. In section 1.2, we
presented the sociogram as the first systematic visualization of a social network.
It was the sociometrists’ main tool to explore and understand the structure of
relations in human groups. In books on graph theory, visualizations are used to
illustrate concepts and proofs. Visualizations facilitate an intuitive understanding
of network concepts, so we will use them a lot.
Our eyes are easily fooled, however. A network can be drawn in many ways
and each drawing stresses other structural features. Therefore, the analyst should
be careful and rely on systematic rather than ad hoc principles for network
drawing. In general, we should use automatic procedures which generate an
optimal layout of the network when we want to explore network structure.
Subsequently, we may edit the automatically generated layout manually if we
want to present it.
Some basic principles of network drawing should be observed. The most
important principle states that the distance between vertices should express the
strength or number of their relations as closely as possible. In a map, the distance
between cities matches their geographical distance. In psychological charts,
spatial proximity of objects usually expresses perceived similarity. Since social
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Exploratory Social Network Analysis with Pajek
 W. de Nooy, A. Mrvar, V. Batagelj
network analysis focuses on relations, a drawing should position vertices
according to their relations: vertices which are connected should be drawn closer
together than vertices which are not related. A good drawing minimizes the
variation in the length of lines. In the case of lines with unequal values, line
length should be proportional to line value.
The legibility of a drawing poses additional demands, which are known as
graph drawing esthetics. Vertices or lines should not be drawn too closely
together and small angles between lines which are incident with the same vertex
should be avoided. Distinct vertices and distinct lines should not merge into one
lump. Vertices should not be drawn on top of a line which belongs to other
vertices. The number of crossing lines should be minimized because the eye tends
to see crossings as vertices.
Application
Draw screen
Pajek offers many ways to draw a network. It has a separate window for drawing,
which is accessed from the Draw>Draw menu in the Main screen. The Draw
screen has an elaborate menu of choices (see Figure 8), some of which will be
presented in later chapters. Use the index of Pajek commands in this book to find
them. We will discuss the most important commands now to help you draw your
first network.
Figure 8 - Draw screen in Pajek.
1.3.4.1
Automatic drawing
Automated procedures for finding an optimal layout are a better way to obtain a
basic layout than manual drawing, because the resulting picture depends less on
the preconceptions and misconceptions of the investigator. Besides, automated
14
Exploratory Social Network Analysis with Pajek
Layout>Energy menu
Layout>Energy>
Starting positions:
random,
circular,
Given xy,
Given z
 W. de Nooy, A. Mrvar, V. Batagelj
drawing is much faster and quite spectacular because the drawing evolves before
your eyes.
In Pajek, several commands for automatic layout are implemented. Two
commands are accessible from the Layout>Energy menu and we will refer to
them as energy commands. Both commands move vertices to locations that
minimize the variation in line length. Imagine that the lines are springs which pull
vertices together. The energy commands ‘pull’ vertices to better positions until
they are in a state of equilibrium. Therefore, these procedures are known as
spring embedders.
The relocation technique used in both automatic layout commands has some
limitations. First, the results depend on the starting positions of vertices. Different
starting positions may yield different results. Most results will be quite similar,
but results can differ markedly. The Starting positions submenu on the
Layout>Energy menu gives you control over the starting positions. You can
choose random and circular starting positions, or you can use the present
positions of vertices as their starting positions: option Given xy, where x and y
refer to the (horizontal and vertical) coordinates of vertices in the plane of the
Draw screen. The fourth option (Given z) refers to the third dimension, which we
will present in Chapter 5.
Figure 9 - Continue dialog box.
Layout>Energy
>Kamada-Kawai>Free,
Fix first and last,
Fix one in the middle
The second limitation of the relocation technique is that it stops if improvement is
relatively small or if the user so requests in a dialog box (Figure 9). This means
that automatic layout generation outputs a drawing that is very good but not
perfect. Manual improvements can be made. However, make small adjustments
only to reduce the risk of discovering a pattern which you introduced to the
drawing yourself. It is usually worth the effort to repeat an energy command a
couple of times with given starting positions to improve a drawing.
Because of these limitations, take the following advice to heart: never rely on
one run of an energy command. Experiment with both commands and manual
adjustment until you obtain the same layout time and again.
The first energy command is named Kamada-Kawai after its authors. This
command produces regularly spaced results, especially for connected networks
which are not very large. It seems to produce more stable results than the other
energy command (Fruchterman Reingold) but it is much slower and it should not
be applied to networks which contain more than 500 vertices. With KamadaKawai, commands Fix first and last and Fix one in the middle enable you to fix
vertices that should not be relocated. The first command fixes the two vertices
with lowest and highest serial numbers. This is very useful if these vertices
15
Exploratory Social Network Analysis with Pajek
Layout>Energy
>Kamada-Kawai
>Selected group only
Layout>Energy
>Fruchterman Reingold
>2D, 3D
Layout>Energy
>Fruchterman Reingold
>Factor
represent the sender and target in an information network. The second command
allows you to specify one vertex that must be placed in the middle of the drawing.
The Kamada-Kawai command can also be applied to a section of a network.
First, use your right mouse button (see above) to zoom in on the section which
you want to energize. Second, select the only Kamada-Kawai submenu command
visible after zooming in, which reads Selected group only. After relocation of the
selected vertices, Pajek asks you whether it should resize the optimized section to
the original network. Yes is usually the right answer to this question.
The second energy command, which is called Fruchterman Reingold, is faster
and it also works with larger networks. This command separates unconnected
parts of the network nicely, whereas Kamada-Kawai draws them on top of one
another. It can generate two and three dimensional layouts as indicated by
commands 2D and 3D in the submenu. We will postpone discussion of three
dimensional visualizations until Chapter 5.
The third command in the submenu, which is labeled Factor, allows the user
to specify the optimal distance between vertices in a drawing which is being
energized with Fruchterman Reingold. This command displays a dialog box
asking for a positive number. A low number yields small distances between
vertices and many vertices are placed in the center of the plane. A high number
pushes vertices out of the center towards positions on a circle. An optimal
distance of 1 is a good starting point. Try a smaller distance if the center of the
drawing is quite empty, but try a higher distance if the center is too crowded.
Each energy command has strong and weak points. We advise starting with
Fruchterman Reingold and using several optimal distances until a stable result
appears. The drawing can then be improved using Kamada-Kawai with the actual
location of vertices as starting positions. Finally, improve the drawing by manual
editing, which we will discuss in the following section.
1.3.4.2
Move>Fix,
Grid,
Circles
 W. de Nooy, A. Mrvar, V. Batagelj
Manual drawing
Pajek supports manual drawing of a network. Use the mouse to drag individual
vertices from one position to another. Place the cursor on a vertex, hold down the
left mouse button, and move the mouse to drag a vertex. As you will see, the lines
which are incident with the vertex move too. You can drop the vertex any place
you want, unless you constrain the movement of vertices with the options in the
Move menu.
The Move menu in the Draw screen has (at least) three options: Fix, Grid, and
Circles. The Fix option allows you to restrict the movement of a vertex. It cannot
be moved horizontally if you select x in the drop down menu, or vertically (select
y). Select Radius to restrict the movement of a vertex to a circle. The options Grid
and Circles allow you to specify a limited number of positions to which a vertex
can be moved. In the case of small networks, this generates esthetically pleasing
results. In Pajek, when you select an option, it is marked by a dot and it is
effective until it is selected again (Figure 10).
16
Exploratory Social Network Analysis with Pajek
 W. de Nooy, A. Mrvar, V. Batagelj
Figure 10 - A selected option in the Draw screen.
Redraw
Previous
Next
GraphOnly
You can zoom in on the drawing by pressing the right mouse button and by
dragging a rectangle over the area which you want to enlarge. Within the
enlargement, you can zoom in again. To return to the entire network, you must
select the Redraw command on the Draw screen menu. It is not possible to zoom
out in steps. The GraphOnly menu is similar to the Redraw menu, except that
GraphOnly removes all labels as well as the heads of the arcs. It shows vertices
and lines only. This option speeds up automatic layout of a network, which is
particularly helpful for drawing large networks. The Previous and Next
commands on the menu allow you to display the network before or after the
current network in the Network drop list, so you need not return to the Main
screen first to select another network. Note, however, that this is true only if the
option Network is selected in the Previous/Next>Apply to submenu of the
Options menu.
Figure 11 - Options menu of the Draw screen.
Options menu:
Transform,
Layout,
Mark vertices using
[Draw]Info
The Options menu offers a variety of choices for changing the appearance of a
network in the Draw screen and for setting options to commands in other Draw
screen menus (Figure 11). Many drawing options are self-explanatory. Options
for changing the shape of the network are listed in the Transform and Layout
submenu, whereas options for the size, color, and labeling of vertices and lines
are found in several other submenus. Figure 11 shows the Options>Mark vertices
using submenu and its options for changing the type of vertex label displayed.
Note the keyboard shortcuts in this submenu: pressing the control key along with
the L, N, or D key has the same effect as selecting the accompanying item in the
submenu.
To improve your drawing, Pajek can evaluate a series of esthetic properties,
such as closest vertices which are not linked, or the number of crossing lines.
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Exploratory Social Network Analysis with Pajek
 W. de Nooy, A. Mrvar, V. Batagelj
This allows you to find the worst aspects of a drawing and to improve them
manually. The Info menu in the Draw screen allows you to select one or all
esthetic properties. If you select one of these commands, Pajek identifies the
vertices which perform worst: they are identified by a color (other than light blue)
in the drawing and they are listed in the Report screen. When you select all
esthetic properties, each property will be associated with a color (see Figure 12
for esthetic information on the dining-table partners network as drawn in Figure
8). Sometimes, vertices violate several esthetic indices, so they ought to be
marked by several colors. Since this is not possible, some colors may not show up
in the drawing.
------------Layout Info
------------YELLOW: The closest vertices: 6 and 15. Distance: 0.07873
GREEN: The smallest angle: 2.1.2. Angle: 0.00000
RED: The shortest line: 9.14. Length: 0.09809
BLUE: The longest line: 11.20. Length: 0.40417
MAGENTA: Number of crossings: 13
WHITE: Closest vertex to line: 6 to 15.25. Distance: 0.02139
Figure 12 - Textual output from Info>Check.
1.3.4.3
[Main]File>Network
>Save
Export menu
Export>Bitmap
Saving a drawing
In order to present pictures of networks to an audience, we have to save our
visualizations. This subsection sketches the options for exporting network
drawings from Pajek.
If a researcher wants to save a layout for future use, the easiest thing to do is
to save the network itself. Remember that a network consists of lists of vertices,
arcs, and edges. The list of vertices specifies a serial number, a label, and
coordinates in the plane for each vertex. Relocation of a vertex in the Draw
screen changes its coordinates and the latest coordinates are written to the
network file on execution of the Save command. When the network is reopened in
Pajek, the researcher obtains the layout s/he saved, that is to say, the general
pattern because the size and colors of vertices and lines are usually not specified
in the network data file.
Pajek offers several options to save the drawing as a picture for presentation
to an audience. The options are listed in the Export menu of the Draw screen.
Three options produce two-dimensional output (EPS/PS, SVG, and Bitmap) and
another three options yield three-dimensional output (VRML, MDL MOLfile,
and Kinemages). Although three-dimensional representations can be quite
spectacular (Figure 13), we will not discuss them now. We will postpone the third
dimension until Chapter 5. We will merely sketch the commands for twodimensional output and we refer to Appendix 2 for more details and additional
viewers.
The Bitmap export option produces an image of the Draw screen: each screen
pixel is represented by a point in a raster, which is called a bitmap. You get
exactly what you see, even the size of the picture matches the size of the Draw
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Exploratory Social Network Analysis with Pajek
 W. de Nooy, A. Mrvar, V. Batagelj
screen. Unfortunately, this means loops are invisible and that reciprocal choices
are represented by one line with two arrowheads rather than two separate arcs.
Every word processor and presentation program operating under Windows can
load and display bitmaps. Bitmaps, however, are cumbersome to edit and they
lose their sharpness if they are enlarged or reduced, e.g., see Figure 8.
Figure 13 - A 3D rendering of the dining-table partners network.
Export>EPS/PS',
SVG,
Options
Vector graphics produce more pleasing results than bitmaps. In a vector graphic,
the shape and position of each circle representing a vertex, each line, arc, loop, or
label is specified. Because each element in the drawing is defined separately, a
picture can be enlarged or reduced without loss of quality and its layout can be
manipulated easily. Pajek can export vector graphics in two formats, which are
closely related: PostScript (command EPS/PS) and Scalable Vector Graphics
(command SVG). PostScript (PS) and Encapsulated PostScript (EPS) are meant
for printing, whereas the Scalar Vector Graphics format is developed for the web.
The user can modify the layout of both PostScript and Scalar Vector Graphics
drawings in the Export>Options submenu, which is covered in detail in Appendix
2.
In this book, we use vector graphics because of their high quality. For
example, the sociogram shown in Figure 1 was exported from Pajek as
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Exploratory Social Network Analysis with Pajek
 W. de Nooy, A. Mrvar, V. Batagelj
Encapsulated PostScript. However, PostScript and Scalar Vector Graphics are
formats that are not universally supported by Windows software. For example,
MS Word 97 cannot display Encapsulated PostScript. Appropriate software, e.g.,
Adobe Illustrator or CorelDraw, is necessary to edit vertices, labels, and lines, or
to convert a vector graphic to a format which your word processor can handle
(e.g., Windows MetaFile). To display Scalar Vector Graphics in a web-page, a
plug-in is needed which can be downloaded from the website of Adobe Systems
Incorporated (see Appendix 2 for details).
1.4
Assembling a social network
In order to perform network analysis, social relations must be measured and
coded. In this section, we will briefly discuss data collection techniques for social
network analysis and explain how to convert data to a network file for Pajek.
There are several ways to collect data on social relations. Traditionally,
sociometrists focus on the structure of social choice within a group. They gather
data by asking each member of a group to indicate his or her favorites (or
opponents) with respect to an activity that is important to the group. For example,
they ask pupils in a class to name the children whom they prefer to sit next to. In
a questionnaire, respondents may write down the names of the children they
choose, or check their names in a list. These methods are called free recall and
roster respectively. The latter method reduces the risk that respondents may
overlook people.
Sometimes, the respondent is asked to nominate a fixed number of favorites.
In sociometry, it was very popular to restrict the number of choices to three. For
example, in the dining-table partners network each girl was asked to make three
choices. This restriction is motivated by the empirical discovery that the more
choices that are allowed the more they concentrate on people who are already
highly chosen. When asked for their best friends, most people mention four
people or less. If they have to mention more people, they usually nominate people
whom they think they should like because they are liked by many others.
However, restricting the number of choices reduces the reliability of the data:
choices are less stable over time and correlate less well than other measurement
techniques such as unrestricted choices, ranking (rank all other group members
with respect to their attractiveness), or paired comparison (list all possible pairs
of group members and choose a preferred person in each pair). A researcher who
fixes the number of choices available to respondents eliminates the difference
between a respondent who entertains many friendships and a loner.
Fixed and free choice, ranking, and paired comparison are techniques that
elicit data on social relations through questioning. However, there are several
data collection techniques that register social relations rather than elicit them. For
example, the amount of interaction between pupils in a class may be observed by
a researcher. Respondents may be asked to register their contacts in a diary,
membership lists and files that log contacts in electronic networks may be coded,
family relations or transactions can be retrieved from archives and databases, et
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Exploratory Social Network Analysis with Pajek
 W. de Nooy, A. Mrvar, V. Batagelj
cetera. The rapid growth of electronic data storage offers new opportunities to
gather data on large social networks.
Indirect data are usually better than reported data, which rely on the often
inaccurate recollections of respondents. However, it is not always easy to identify
people and organizations unambiguously in data collected indirectly: is a Mr.
Jones on the board of one organization the same person as the Mr. Jones who is
CEO of another firm? It goes without saying that network analysis demands
correct identification of vertices in the network.
Application
Now that you have collected your data, it is time to create a network that can be
analyzed with Pajek. In Section 1.3.1, we discussed the structure of a Pajek
network file. This is a simple text file, which can be typed out in any word
processor that exports plain text. Do not forget to attribute serial numbers to the
vertices ranging from 1 to the number of vertices. Save the network from the
word processor as plain, unformatted text (DOS text, ASCII) and use the
extension .net in the file name.
Figure 14 - Random network without lines.
Net>Random Network
>Total No. of Arcs
File>Network>Edit
Editing Network screen
It is also possible to produce the network file in Pajek. First, make a new random
network with the command Net>Random Network>Total No. of Arcs. In the first
dialog box, type in the number of vertices you want. In the second dialog box,
request zero arcs to obtain a network without lines. The command creates a new
network and adds it to the Network drop list in the Main screen. When you draw
it (Draw>Draw), you will see that the vertices are nicely arranged in a circle
(Figure 14).
As a second step, add lines to the network, which may be done in the Main
screen or in the Draw screen. In the Main screen, both the Edit button at the side
21
Exploratory Social Network Analysis with Pajek
[Editing Network]
Newline
 W. de Nooy, A. Mrvar, V. Batagelj
of the Network drop list (a picture of a writing hand) and the command
File>Network>Edit open a dialog box which allows you to select a vertex by
serial number or by label. Next, the Editing Network screen is shown for the
selected vertex (Figure 15). In the Draw screen, you can open the Editing
Network screen by right-clicking a vertex.
Double-clicking the word Newline in the Editing Network screen opens a
dialog box which allows the user to add a line to or from the selected vertex. To
add an edge type the number of another vertex. Type the number of the vertex
preceded by a + sign to add an arc to the selected vertex and use a - sign to add an
arc from the selected vertex. Each new edge and arc is displayed as a line in the
Editing Network screen. For example, Figure 15 displays an arc from vertex 4 to
vertex 1, an arc from vertex 1 to 3, and an edge (indicated by a dash between
vertex numbers) between vertices 1 and 2. Also, you can delete a line in the
Editing Network screen: just double-click the line you want to delete.
Figure 15 - Edit Network screen.
File>Network>Save
By default, all lines have unit line value as indicated by the expression val=1.000
in the Editing Network screen. Line values can be changed in this screen by
selecting the line (left-click) and right-clicking it. A dialog box appears that
accepts any number (positive, zero and negative) as input.
As a final step, save the network. Networks in Pajek are not automatically
saved. Because network analysis usually yields many new networks, most of
which are just intermediate steps, Pajek does not prompt the user to save
networks. Save the new network as soon as you finish editing it. Use Save from
the submenu File>Network or use the save button (the picture of a diskette) at the
side of the Network drop list. We recommend that you save a network in Pajek
Arcs/Edges format for easy manual editing and for a maximum choice of layout
options (see Appendix 2). Give the file a meaningful name and the extension .net.
It is possible to generate ready-to-use network files from spreadsheets and
databases by exporting the relevant data in plain text format. For medium or large
networks especially, processing the data as a spreadsheet or relational database
helps data cleaning and coding. We will, however, not go into details here.
1.5
Summary
This chapter introduces social network analysis and emphasizes its theoretical
interest in social relations between people or organizations, and its roots in
mathematical graph theory. A network is defined as a set of vertices and a set of
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Exploratory Social Network Analysis with Pajek
 W. de Nooy, A. Mrvar, V. Batagelj
lines between vertices with additional information on the vertices or lines. This
flexible definition permits a wide variety of empirical phenomena, ranging from
the structure of molecules to the structure of the universe, to be modeled as
networks. In social network analysis, we concentrate on relations between people
or social entities that represent people, e.g. affective relations between people,
trade relations between organizations, or power relations between nations.
This simple definition of a network covers all types of networks which we
will encounter in this book: directed and undirected networks, networks with and
without loops or multiple lines. Most social networks are simple undirected or
directed networks, which do not contain multiple lines, and loops usually do not
occur. As a result of transformations, however, networks may acquire multiple
lines and loops. The results of analytic procedures may depend on the kind of
network we are analyzing, so it is important to know what kind of network it is.
The mathematical roots of network analysis permits powerful and welldefined manipulations, calculations, and visualizations of social networks.
Exploratory social network analysis as we present it here makes ample use of
these three techniques. In exploring a social network, we will first visualize it to
get an impression of its structure. The sociogram, which originates from
sociometry, is our main visualization technique.
We are convinced that doing social network analysis is a good way of
learning about it. Therefore, the program Pajek for network analysis is an integral
part of the course. We urge you to practice the commands demonstrated rather
than just read about them. The data related to each example in the book are
supplied on the CD ROM. This chapter introduces the menu structure of Pajek
and some basic commands for visualizing networks. You are now ready to start
analyzing social networks.
1.6
1
2
Exercises
A sociogram is:
a an index of sociability
b a graphic representation of group structure
c the structure of sociometric choices
d a set of vertices connected by edges or arcs
Which of the following definitions is correct?
a a graph is a set of vertices and a set of lines
b a graph is a set of vertices and a set of edges
c a graph is a network
d a graph is a set of vertices and a set of pairs of vertices
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Exploratory Social Network Analysis with Pajek
3
Have a look at the networks depicted below and choose the correct statement.
A
4
5
6
7
8
 W. de Nooy, A. Mrvar, V. Batagelj
B
a neither A nor B is a simple directed graph
b A is not a simple directed graph but B is
c A and B are simple directed graphs
d A and B are networks but not graphs
Social network analysis is exploratory if:
a the researcher has no specific ideas on the structure of the network
beforehand
b it deals with social settings that are unexplored by social scientists
c the network is studied outside a laboratory
d it does not try to predict network structure from a sample
Which of the following statements is correct?
a a line can be incident with a line
b a line can be incident with a vertex
c an edge can be incident with a line
d a vertex can be incident with a vertex
Open the dining-table partners network and improve it according to the
esthetic criterion for minimizing crossing lines. Which vertex can you move
to reduce the number of crossing lines?
Open the dining-table partners network and remove all second choices with
the command Net>Transform>Remove>lines with value>higher than.
a Why is this command part of the Net menu?
b What in your opinion is the most striking result, if you draw the new
network?
c Which energy command do you recommend to optimize this drawing?
Explain your choice.
Use the Fruchterman Reingold command with random starting positions and
its Factor option to get a sociogram from the dining-table partners network
with a clear distinction between vertices in the center and vertices in the
margin or periphery. Do you think your drawing is better than the original?
Justify your answer.
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Exploratory Social Network Analysis with Pajek
1.7
 W. de Nooy, A. Mrvar, V. Batagelj
Assignment
Your first social network analysis: investigate the friendship network within your
class as they are or as you perceive them. Choose the right kind of network to
conceptualize friendship relations (directed, undirected, valued, with multiple
lines and loops?), collect the data (design a questionnaire or state the friendships
which you observe), make a Pajek network file, draw and interpret it. If you use
perceived friendship relations, it is worthwhile comparing your network to
networks made by other students. Where do they differ?
1.8
•
•
•
1.9
1
2
3
4
5
6
Further Reading
The example of the girls’ school dormitory is taken from J.L. Moreno, The
Sociometry Reader. Glencoe (Ill.), The Free Press, 1960, p. 35.
A brief history of social network analysis can be found in J. Scott, Social
Network Analysis. A Handbook. London, Sage Publications, 1991 (2nd ed.
2000), Chapter 2.
For the problem of boundary specification and sampling see S. Wasserman &
K. Faust, Social Network Analysis: Methods and Applications. Cambridge,
Cambridge University Press, section 2.2, p. 30-35. Information on data
collection can be found in section 2.4 (p. 43-59). This impressive monograph
is a good starting point for further reading on any topic in social network
analysis.
Answers to questions
Answer b is correct because a sociogram is a specific way of drawing the
structure of relations within a human group.
A graph is defined as a set of vertices and a set of lines between pairs of
vertices (see Section 1.3.1), so answer d is correct: lines can be defined as
pairs of vertices. Answer a does not specify that the lines must have the
graph’s vertices as their endpoints, so this answer is not correct. Answer b
has the same fault and it restricts the definition to undirected graphs, which is
not correct. Answer c ignores the difference between a graph and a network.
Answer b is correct. A simple directed graph contains at least one arc (so it is
directed) and no multiple lines; it may contain loops. Therefore, network B is
a simple directed graph but network A is not due to its multiple lines. Answer
d is wrong because there is no additional information on the vertices and
arcs, such as vertex labels or line values.
Answers a and d are correct.
Answer b is correct.
Select option No. of crossings in the Info menu of the Draw screen. The
vertices in the center are colored pink now: their lines cross. The Report
screen shows the number of crossing lines to be 13. The easiest improvement
is to drag Alice to a position above the arc from Laura to Eva. Now, Alice’s
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Exploratory Social Network Analysis with Pajek
7
8
 W. de Nooy, A. Mrvar, V. Batagelj
arcs do not cross any other and the number of crossing lines is reduced to 12.
However, it is even better to drag Alice and Martha to the left of the arc from
Francis to Eva, because Martha and Marion are connected by mutual choice,
which counts twice at every crossing. If you do this, it is better to drag
Maxine below the arc from Martha to Anna, because Eva and Maxine are
doubly connected. Another improvement is made if Adele is placed to the
left of the arc from Anna to Lena. We reach a minimum of 7 crossing arcs.
Can you do better?
Note: since first choices have line values 1 and second choices have line
values 2, you must type 1 in the dialog box which appears on activation of
the command Net>Transform>Remove>lines with value>higher than.
a This command operates on one object, namely a single network.
b The network is no longer connected. Some girls and pairs of girls are
disconnected, e.g., Hazel, Cora & Ada, and Jean & Helen.
c We prefer Fruchterman Reingold because Kamada-Kawai sometimes
draws disconnected parts on top of one another.
Setting Factor to 2 or higher will yield a drawing with a lot of vertices placed
on a circle and few vertices in the middle, notably Anna, Eva, or Edna. This
drawing is not an improvement, the selection of central vertices is arbitrary.
Repeat the energy command several times (with random starting positions)
and you will find different persons in the center. Also, several esthetic
criteria are violated more than in the original drawing.
26